Stability of the recursive sequence \(x_{n+1}=(\alpha-\beta x_n)/(\gamma+x_{n-1})\) (Q5946935)

Consider the recursive sequence NEWLINE\[NEWLINEx_{n+1}= {\alpha+ \beta x_n \over \gamma+x_{n-1}},\;n=0,1,\dots \tag{*}NEWLINE\]NEWLINE where \(\alpha,\beta\) and \(\gamma\) are nonnegative and the initial conditions \(x_1\) and \(x_0\) are arbitrary. Equation (*) has two equilibrium points positive and negative.NEWLINENEWLINENEWLINEIf there exists \(k\geq 2\) such that \(\gamma\geq k\alpha/ \beta\) and \(\alpha\geq k\beta^2\), then the positive equilibrium point is a global attractor with some given basin. The asymptotic properties in the case \(\alpha=0\), \(\beta<0\), \(\gamma >0\) are investigated in details.